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LRCS: Duality, LP bounds, and field size.

Anina Gruica1, Benjamin Jany2, Alberto Ravagnani2

  • 1Technical University of Denmark, Kongens Lyngby, Denmark.

Designs, Codes, and Cryptography
|April 28, 2026
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Summary

This study introduces a duality theory for locally recoverable codes (LRCs), establishing new parameter bounds. It reveals that optimal LRCs require small minimum distance and block length relative to the field size.

Keywords:
Distributed storageError-correcting codesLP boundLocally recoverable codes

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Area of Science:

  • Coding Theory
  • Information Theory
  • Discrete Mathematics

Background:

  • Locally Recoverable Codes (LRCs) are essential for distributed storage systems, offering fault tolerance.
  • Existing bounds on LRC parameters do not fully capture their locality properties.
  • The MacWilliams identity is a fundamental tool in coding theory, relating code properties to their duals.

Purpose of the Study:

  • To develop a duality theory for locally recoverable codes (LRCs).
  • To establish new and improved bounds on the parameters of LRCs.
  • To investigate the existence of optimal LRCs over small fields.

Main Methods:

  • Development of a duality theory for LRCs.
  • Introduction of a refined weight distribution capturing locality.
  • Application of a MacWilliams-type identity for deriving LP-type bounds.
  • Utilizing dual distance bounds and generalized weights.

Main Results:

  • A new duality theory for LRCs is established.
  • An LP-type bound is derived, outperforming existing bounds in certain cases.
  • Non-existence results for optimal LRCs over small fields are obtained.
  • It is shown that optimal LRCs necessitate small minimum distance and block length relative to field size.

Conclusions:

  • The developed duality theory provides a powerful framework for analyzing LRCs.
  • The new bounds offer tighter constraints on LRC parameters.
  • The findings have implications for the design of efficient and robust error-correcting codes for storage systems.